Past exam of the stochastic differential equations course of the University of Oxford / 2011 / 2

The given Doob L2 maximal inequality givesApply Markov's inequality to the nonnegative variable $(M_t^{\ast }{)}^2$, at the threshold printed in the question:The corresponding threshold for $M_t^{\ast }$ itself is $2^{-(n-1)/2}$.

Set $A_n=\{{\sup }_{s\le t}|M_s^{(n)}|\ge 2^{-(n-1)/2}\}$. Part (a) gives ${\sum }_n\mathbb{P}(A_n)<\infty$. The Borel-Cantelli first lemma implies that, almost surely, only finitely many $A_n$ occur. On the complementary full-probability event, all sufficiently late terms satisfyThese deterministic bounds form a convergent geometric series. The finitely many earlier suprema are finite by continuity on the compact time interval. HenceThe series therefore converges absolutely and uniformly on $[0,t]$ by the Weierstrass M-test, and its sample-path limit is continuous.

Work modulo indistinguishability of stochastic processes, with norm $\Vert M\Vert =(\mathbb{E}|M_t{|}^2{)}^{1/2}$. It is a genuine norm: zero terminal value implies $M_s=\mathbb{E}[M_t\mid {\mathcal{F}}_s]=0$ at every fixed time, and continuity extends equality from countably many rational times to all times simultaneously.

Let $M^j$ be a Cauchy sequence in this norm. Choose a subsequence with $\Vert M^{j_{n+1}}-M^{j_n}\Vert <2^{-n}$. Part (b) applies to these difference martingales, so their telescoping series converges uniformly almost surely. Its limit has continuous paths. At each time $s$, conditional-expectation contractivity givesso the fixed-time limit also exists in $L^2$ and agrees with the pathwise limit. It has an adapted version obtained by taking the finite pointwise subsequence limit when it exists, and zero otherwise. Each such definition is measurable at its own time, and the almost surely uniform limit gives continuous paths. Passing to the $L^2$ limit in $\mathbb{E}[M_v^j\mid {\mathcal{F}}_s]=M_s^j$ shows that the limit is a martingale. The original Cauchy sequence converges to it in terminal norm. Thus the space is complete.

Part (b) supplies what fixed-time $L^2$ completeness alone does not: a single full-probability set on which a subsequence converges uniformly in time, preserving path continuity. This is completeness of continuous square-integrable martingales.

For simple predictable integrands, the Itô integral is a continuous square-integrable martingale and the Itô isometry givesApproximate a general square-integrable predictable integrand by simple ones. The resulting integral martingales are Cauchy in terminal $L^2$ norm. Completeness of continuous square-integrable martingales guarantees a unique continuous martingale limit, independent of the approximation, which defines the integral on the completed integrand space.